![](/files/happy5.png)
Wall-Sun-Sun primes and Fibonacci divisibility
Conjecture For any prime
, there exists a Fibonacci number divisible by
exactly once.
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
Equivalently:
Conjecture For any prime
,
does not divide
where
is the Legendre symbol.
![$ p>5 $](/files/tex/a9737aaa01b2918b7e8ab9f12ab0a7c463381427.png)
![$ p^2 $](/files/tex/8f6ab14c3cc11c005c9b97778062620c197ff66d.png)
![$ F_{p-\left(\frac p5\right)} $](/files/tex/e1a04059df8fc3e49db781f306092d0d564732e3.png)
![$ \left(\frac mn\right) $](/files/tex/d8d0b3deced8bb07e56b3da3337f2d8c7db060a1.png)
Let be an odd prime, and let
denote the
-adic valuation of
. Let
be the smallest Fibonacci number that is divisible by
(which must exist by a simple counting argument). A well-known result says that
unless
divides
, and
. This conjecture asserts that
for all
. This has been verified up to at least
. [EJ]
This conjecture is equivalent to non-existence of Wall-Sun-Sun primes.
Bibliography
[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, The Fibonacci sequence modulo p^2
[R] Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli
*[W] D. D. Wall, Fibonacci Series Modulo m, American Mathematical Monthly, 67 (1960), pp. 525-532.
* indicates original appearance(s) of problem.